Earth models are utilized in the petroleum industry to understand the nature of a particular subsurface reservoir. Earth models provide a numerical representation of a reservoir property as a function of location and are constructed in the form of a single geological representation. Earth models are typically constrained or shaped by empirical data of the reservoir, such as seismic data, geologic data, drilling data, and production data. Geoscientists typically construct a plurality of earth models using stochastic techniques, such that the earth models represent extremes in reservoir porosity, water saturation, and permeability. The individual models can be analyzed to evaluate the geological uncertainty of the subsurface reservoir. For example, the earth models can be simulated under various operating scenarios to forecast the hydrocarbon production of the subsurface reservoir and the simulation runs can be interpreted and analyzed to obtain simple fluid flow characteristics of the subsurface reservoir. For instance, various well patterns can be implemented to see how they impact the forecasted production. Such evaluation is typically performed by determining static measures of heterogeneity for a given model.
Static measures of heterogeneity concentrate on the level of permeability variation in a reservoir. To calculate static measures of heterogeneity for a given model, Dykstra-Parsons and Lorenz coefficients can be calculated. These coefficients are typically derived from a Lorenz plot constructed from the model's permeability, layer thickness, and porosity distributions. Simple flow geometries can be determined for a reservoir by generating flow capacity-storage capacity curves, which are based on static data. There are many methods known in the art for plotting flow capacity-storage capacity curves, which are also commonly referred to as F-C curves or F-Φ curves.
As an example, flow capacity-storage capacity curves can be constructed for individual flow paths within a layered reservoir. In this case, the flow paths are represented as layers that have unique values of permeability, porosity, cross sectional area, and length. The flow capacity of an individual streamline can be described as the volumetric flow of that layer, divided by the total volumetric flow. Therefore, the flow capacity fi can be computed using Darcy's law and defining N layers each having a different permeability k, porosity φ, and thickness h using the following equation:
                              f          i                =                                            q              i                                                      ∑                                  i                  =                  1                                N                            ⁢                              q                i                                              =                                                    (                kh                )                            i                                                      ∑                                  i                  =                  1                                N                            ⁢                                                (                  kh                  )                                i                                                                        (                  Equation          ⁢                                          ⁢          1                )            Similarly, the storage capacity of layer “i” can be computed as the layer pore volume divided by the total pore volume:
                              c          i                =                                            V              Pi                                                      ∑                                  i                  =                  1                                N                            ⁢                              V                Pi                                              =                                                    (                                  φ                  ⁢                                                                          ⁢                  h                                )                            i                                                      ∑                                  i                  =                  1                                N                            ⁢                                                (                                      φ                    ⁢                                                                                  ⁢                    h                                    )                                i                                                                        (                  Equation          ⁢                                          ⁢          2                )            
A F-C diagram is constructed by computing the cumulate distribution function of ƒ and c. Therefore, the cumulative distribution functions for Fi, which represents the volumetric flow of all layers, and for Ci, which represents the pore volume associated with those layers, can be written as:
                              F          i                =                                                            ∑                                  j                  =                  1                                i                            ⁢                              q                j                                                                    ∑                                  j                  =                  1                                N                            ⁢                              q                i                                              =                                                    ∑                                  j                  =                  1                                i                            ⁢                                                (                  kh                  )                                j                                                                    ∑                                  j                  =                  1                                N                            ⁢                                                (                  kh                  )                                j                                                                        (                  Equation          ⁢                                          ⁢          3                )                                          C          i                =                                                            ∑                                  j                  =                  1                                i                            ⁢                              Vp                j                                                                    ∑                                  j                  =                  1                                N                            ⁢                              Vp                j                                              =                                                    ∑                                  j                  =                  1                                i                            ⁢                                                (                                      φ                    ⁢                                                                                  ⁢                    h                                    )                                j                                                                    ∑                                  j                  =                  1                                N                            ⁢                                                (                                      φ                    ⁢                                                                                  ⁢                    h                                    )                                j                                                                        (                  Equation          ⁢                                          ⁢          4                )            Calculations using Equations 1-4 for a simple 5-layer model are provided as an example:
h (ft)k (md)φkhφhFC0055000.2525001.250.75640.33351000.250010.90770.65500.152500.750.98340.85100.1500.50.99850.933510.0550.2511Σ33053.75
The resulting F-C curve for this example is given in FIG. 1. Calculating the F-C curve from static data or by assuming a simple flow geometry, as we have in the present example, is relatively straightforward. However, this analysis does not take into account the possibility of a variable flow path length, which is common in heterogeneous media. Measures of static heterogeneity are unable to capture how fluid flow is impacted by connectivity between a production well and a fluid injection well. For example, a low permeability, short path typically cannot be differentiated from a high permeability, long path because both flow paths have a similar residence time.
Therefore, while measures of static heterogeneity can be analyzed to describe some aspects of heterogeneity, they cannot describe how “connected” that heterogeneity is. Furthermore, differences in each model's recovery behavior is not quantitatively obtained. For example, there is no guarantee that varying the permeability variance of an earth model will result in changes to the recovery efficiency, which can be described as the percentage of oil recovered. Similarly, this change also will not indicate how the sweep efficiency, which can be described as the percentage of oil recovery verses time, will be impacted. Therefore, while simulation of earth models provides for prediction of reservoir performance, it does not allow for rapid evaluation of how particular earth model characteristics influence the predicted performance. Moreover, there is no present method of evaluating the dynamic heterogeneity of an earth model to unambiguously rank it against other earth models.